Conjugacy-separable implies residually finite

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., conjugacy-separable group) must also satisfy the second group property (i.e., residually finite group)
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Statement

Any conjugacy-separable group is a residually finite group.

Definitions used

Conjugacy-separable group

Further information: Conjugacy-separable group

A group G is termed conjugacy-separable if given any elements x,y \in G that are not conjugate in G, there exists a normal subgroup of finite index N such that the images of x</mah> and <math>y in G/N are not conjugate.

Residually finite group

Further information: Residually finite group

A group G is termed residually finite if given any non-identity element x \in G, there is a normal subgroup of finite index N in G such that x is not in N.

Proof

Given: A conjugacy-separable group G, a non-identity element x \in G.

To prove: There is a normal subgroup of finite index N in G such that x is not in N.

Proof: Let e be the identity element. Since no non-identity element is conjugate to the identity element, x and e are in distinct conjugacy classes. Thus, there exists a normal subgroup N of finite index such that the image of x is not conjugate to the image of e. In particular, this implies that x is not contained in N, so we are done.